###### Advertisements

###### Advertisements

A radioactive element disintegrates for an interval of time equal to its mean lifetime. The fraction that has disintegrated is ______

#### Options

`1/"e"`

`("e" - 1)/"e"`

`"e"/("e" - 1)`

`"e"/2`

###### Advertisements

#### Solution

A radioactive element disintegrates for an interval of time equal to its mean lifetime. The fraction that has disintegrated is `underline(("e" - 1)/"e")`.

**Explanation:**

Mean life time `tau = 1/lambda`

Now, N = `"N"_0"e"^{-lambda"t"}`

∴ For t = τ = `1/lambda`

`"N"/"N"_0 = "e"^{-lambda xx 1/lambda} = 1/"e"`

∴ The fraction that has disintegrated is given as,

`1 - "N"/"N"_0 = 1 - 1/"e" = ("e" - 1)/"e"`

#### RELATED QUESTIONS

The decay constant of radioactive substance is 4.33 x 10^{-4} per year. Calculate its half life period.

(a) Write the basic nuclear process involved in the emission of β^{+} in a symbolic form, by a radioactive nucleus.

(b) In the reactions given below:

(i)`""_16^11C->_y^zB+x+v`

(ii)`""_6^12C+_6^12C->_a^20 Ne + _b^c He`

Find the values of *x, y*, and *z* and *a, b* and* c*.

State the law of radioactive decay.

Derive the mathematical expression for law of radioactive decay for a sample of a radioactive nucleus

How is the mean life of a given radioactive nucleus related to the decay constant?

Why is it found experimentally difficult to detect neutrinos in nuclear β-decay?

A source contains two phosphorous radio nuclides `""_15^32"P"` (T_{1/2 =} 14.3d) and `""_15^33"P"` (T_{1/2} = 25.3d). Initially, 10% of the decays come from `""_15^33"P"`. How long one must wait until 90% do so?

Under certain circumstances, a nucleus can decay by emitting a particle more massive than an α-particle. Consider the following decay processes:

\[\ce{^223_88Ra -> ^209_82Pb + ^14_6C}\]

\[\ce{^223_88 Ra -> ^219_86 Rn + ^4_2He}\]

Calculate the Q-values for these decays and determine that both are energetically allowed.

Represent Radioactive Decay curve using relation `N = N_o e^(-lambdat)` graphically

A radioactive nucleus 'A' undergoes a series of decays as given below:

The mass number and atomic number of A_{2} are 176 and 71 respectively. Determine the mass and atomic numbers of A_{4} and A.

Using the equation `N = N_0e^(-lambdat)` obtain the relation between half-life (T) and decay constant (`lambda`) of a radioactive substance.

The radioactive isotope D decays according to the sequence

If the mass number and atomic number of D_{2} are 176 and 71 respectively, what is (i) the mass number (ii) atomic number of D?

The decay constant of a radioactive sample is λ. The half-life and the average-life of the sample are respectively

Lithium (*Z* = 3) has two stable isotopes ^{6}Li and ^{7}Li. When neutrons are bombarded on lithium sample, electrons and α-particles are ejected. Write down the nuclear process taking place.

The masses of ^{11}C and ^{11}B are respectively 11.0114 u and 11.0093 u. Find the maximum energy a positron can have in the β*-decay of ^{11}C to ^{11}B.

(Use Mass of proton m_{p} = 1.007276 u, Mass of `""_1^1"H"` atom = 1.007825 u, Mass of neutron m_{n} = 1.008665 u, Mass of electron = 0.0005486 u ≈ 511 keV/c^{2},1 u = 931 MeV/c^{2}.)

^{28}Th emits an alpha particle to reduce to ^{224}Ra. Calculate the kinetic energy of the alpha particle emitted in the following decay:

`""^228"Th" → ""^224"Ra"^(∗) + alpha`

`""^224"Ra"^(∗) → ""^224"Ra" + γ (217 "keV")`.

Atomic mass of ^{228}Th is 228.028726 u, that of ^{224}Ra is 224.020196 u and that of `""_2^4H` is 4.00260 u.

(Use Mass of proton m_{p} = 1.007276 u, Mass of `""_1^1"H"` atom = 1.007825 u, Mass of neutron m_{n} = 1.008665 u, Mass of electron = 0.0005486 u ≈ 511 keV/c^{2},1 u = 931 MeV/c^{2}.)

When charcoal is prepared from a living tree, it shows a disintegration rate of 15.3 disintegrations of ^{14}C per gram per minute. A sample from an ancient piece of charcoal shows ^{14}C activity to be 12.3 disintegrations per gram per minute. How old is this sample? Half-life of ^{14}C is 5730 y.

A radioactive isotope is being produced at a constant rate dN/dt = R in an experiment. The isotope has a half-life t_{1}_{/2}. Show that after a time t >> t_{1}_{/2} the number of active nuclei will become constant. Find the value of this constant.

Consider the situation of the previous problem. Suppose the production of the radioactive isotope starts at t = 0. Find the number of active nuclei at time t.

The half-life of ^{40}K is 1.30 × 10^{9} y. A sample of 1.00 g of pure KCI gives 160 counts s^{−1}. Calculate the relative abundance of ^{40}K (fraction of ^{40}K present) in natural potassium.

Obtain a relation between the half-life of a radioactive substance and decay constant (λ).

What is the amount of \[\ce{_27^60Co}\] necessary to provide a radioactive source of strength 10.0 mCi, its half-life being 5.3 years?

Disintegration rate of a sample is 10^{10} per hour at 20 hours from the start. It reduces to 6.3 x 10^{9} per hour after 30 hours. Calculate its half-life and the initial number of radioactive atoms in the sample.

The isotope \[\ce{^57Co}\] decays by electron capture to \[\ce{^57Fe}\] with a half-life of 272 d. The \[\ce{^57Fe}\] nucleus is produced in an excited state, and it almost instantaneously emits gamma rays.

(a) Find the mean lifetime and decay constant for ^{57}Co.

(b) If the activity of a radiation source ^{57}Co is 2.0 µCi now, how many ^{57}Co nuclei does the source contain?

c) What will be the activity after one year?

A source contains two species of phosphorous nuclei, \[\ce{_15^32P}\] (T_{1/2} = 14.3 d) and \[\ce{_15^33P}\] (T_{1/2} = 25.3 d). At time t = 0, 90% of the decays are from \[\ce{_15^32P}\]. How much time has to elapse for only 15% of the decays to be from \[\ce{_15^32P}\]?

Before the year 1900 the activity per unit mass of atmospheric carbon due to the presence of ^{14}C averaged about 0.255 Bq per gram of carbon.

(a) What fraction of carbon atoms were ^{14}C?

(b) An archaeological specimen containing 500 mg of carbon, shows 174 decays in one hour. What is the age of the specimen, assuming that its activity per unit mass of carbon when the specimen died was equal to the average value of the air? The half-life of ^{14}C is 5730 years.

Obtain an expression for the decay law of radioactivity. Hence show that the activity A(t) =λN_{O} e^{-λt}.

Two radioactive materials X_{1} and X_{2} have decay constants 10λ and λ respectively. If initially, they have the same number of nuclei, then the ratio of the number of nuclei of X_{1} to that of X_{2} will belie after a time.

Which one of the following nuclei has shorter meant life?

'Half-life' of a radioactive substance accounts for ______.

The half-life of a radioactive sample undergoing `alpha` - decay is 1.4 x 10^{17} s. If the number of nuclei in the sample is 2.0 x 10^{21}, the activity of the sample is nearly ____________.

After 1 hour, `(1/8)^"th"` of the initial mass of a certain radioactive isotope remains undecayed. The half-life of the isotopes is ______.

Two radioactive materials Y_{1} and Y_{2} have decay constants '5`lambda`' and `lambda` respectively. Initially they have same number of nuclei. After time 't', the ratio of number of nuclei of Y_{1} to that of Y_{2 }is `1/"e"`, then 't' is equal to ______.

What percentage of radioactive substance is left after five half-lives?

Two electrons are ejected in opposite directions from radioactive atoms in a sample of radioactive material. Let c denote the speed of light. Each electron has a speed of 0.67 c as measured by an observer in the laboratory. Their relative velocity is given by ______.

The half-life of a radioactive nuclide is 20 hrs. The fraction of the original activity that will remain after 40 hrs is ______.

If 10% of a radioactive material decay in 5 days, then the amount of original material left after 20 days is approximately :

The half-life of the radioactive substance is 40 days. The substance will disintegrate completely in

Samples of two radioactive nuclides A and B are taken. λ_{A} and λ_{B} are the disintegration constants of A and B respectively. In which of the following cases, the two samples can simultaneously have the same decay rate at any time?

- Initial rate of decay of A is twice the initial rate of decay of B and λ
_{A}= λ_{B}. - Initial rate of decay of A is twice the initial rate of decay of B and λ
_{A}> λ_{B}. - Initial rate of decay of B is twice the initial rate of decay of A and λ
_{A}> λ_{B}. - Initial rate of decay of B is the same as the rate of decay of A at t = 2h and λ
_{B}< λ_{A}.

The radioactivity of an old sample of whisky due to tritium (half-life 12.5 years) was found to be only about 4% of that measured in a recently purchased bottle marked 10 years old. The age of a sample is ______ years.

What is the half-life period of a radioactive material if its activity drops to 1/16^{th} of its initial value of 30 years?

The half-life of `""_82^210Pb` is 22.3 y. How long will it take for its activity 0 30% of the initial activity?